#7891190

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10,14,17)(11,18,15,12)(13,16,21,23)(19,24,22,20)(25,26)(27,32)(28,30)(29,39)(31,35)(33,38)(34,36)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,22)(23,24)(25,27,31,37)(26,32,35,40)(28,36,38,29)(30,34,33,39);(9,12,14,18)(10,15,17,11)(13,20,21,24)(16,22,23,19)(25,28,31,38)(26,33,35,30)(27,29,37,36)(32,34,40,39);(1,3,4,7)(2,5,6,8)(9,12)(10,11)(13,20)(14,18)(15,17)(16,19)(21,24)(22,23)(25,28)(26,30)(27,36)(29,37)(31,38)(32,34)(33,35)(39,40);(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,22)(23,24)(25,29,31,36)(26,34,35,39)(27,28,37,38)(30,32,33,40);(9,13,14,21)(10,16,17,23)(11,19,15,22)(12,20,18,24)(25,30)(26,28)(27,34)(29,40)(31,33)(32,36)(35,38)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=ghi,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7798386);